Journal of Radiology and Imaging
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Journal of Radiology and Imaging
Volume 8, Issue 2, August 2024, Pages 410
Original researchOpen Access
Consideration for a revised Gaussianpencilbeammodel reported for calculation of the inwater dose caused by clinical electronbeam irradiation

Akira Iwasaki^{1,*}, Shingo Terashima^{2}, Shigenobu Kimura^{3}, Kohji Sutoh^{3}, Kazuo Kamimura^{3}, Yoichiro Hosokawa^{2} and
Masanori Miyazawa^{4}
 ^{1 }2324 Shimizu, Hirosaki, Aomori 0368254, Japan
 ^{2 }Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan
 ^{3 }Department of Radiology, Aomori City Hospital, 11420 Katta, Aomori 0300821, Japan
 ^{4 }Technology of Radiotherapy Corporation, 212 Koishikawa, Bunkyoku, Tokyo 1750092, Japan
*Corresponding authors: Akira Iwasaki, 2324 Shimizu, Hirosaki, Aomori 0368254, Japan. Tel.: +172332480; Email: fmcch384@ybb.ne.jp or fmcch384@gmail.com; and Shingo Terashima, Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan. Tel.: +81172395525; Email: stera@hirosakiu.ac.jp
Received 28 March 2024 Revised 20 May 2024 Accepted 30 May 2024 Published 8 June 2024
DOI: http://dx.doi.org/10.14312/23998172.20242
Copyright: © 2020 Iwasaki A, et al. Published by NobleResearch Publishers. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.
AbstractTop
Purposes: We perform further development for our previous Gaussianpencilbeammodel used for calculating the electron dose in water under clinical electronbeam irradiation. The main purpose is to evaluate accurately the parallel beam depthdoses at deep depths beyond about the extrapolated range (Rp) under an infinite field. Methods: Sets of parallel beam depthdoses under an infinite field were reconstructed for the same beams of E=6, 12, and 18 MeV in light of the electron Monte Carlo (eMC) datasets as reported by Wieslander and Knöös (2006), separating the datasets into the direct electron beam and directplusindirect electron beam groups. Particularly, for each electron beam, we took serious views of the depthdose (DD) curves near the beam entrance surface and of the OAD (offaxis dose) curves at deep depths beyond about the extrapolated range (R_{p}) under an infinite field. Results and conclusions: The following results were obtained by comparing the calculated DD and OAD datasets with the eMC datasets: (i) The revised Gaussian pencil beam model is of practical use without using complicated correction factors; and (ii) The DD and OAD datasets are yielded effectively over wide ranges of depths and offaxis distances, respectively.
Keywords: Gaussian pencil beam model; dose calculation; electron MC; electron beams; linear accelerator
Research highlightsTop
The dose caused by the clinical electronbeam irradiation is mainly composed of the doses due to the direct electrons, the indirect electrons, and the contaminant photons. In light of the electron Monte Carlo (eMC) datasets reported by Wieslander and Knöös (2006), the present paper describes further development of the Gaussianpencilbeammodel for calculating doses in a homogeneous water phantom for direct electron beams and directplusindirect electron beams. The study subjects are how to calculate dosedatasets accurately both near the beam entrance surface and at deep depths. Accurate datasets of depthdose (DD) and offaxis dose (OAD) were obtained from the shallow to deep depths for each of the electron beams.
1. IntroductionTop
Wieslander and Knöös [1, 2] have reported characteristic features of the dose in homogeneous water, caused by clinical electronbeam irradiations, using an electron Monte Carlo (eMC) method for 6, 12, and 18MeV electron beams by taking 10×10 cm^{2} and 10×10/14×14 cm^{2} applicators. The 10×10 cm^{2} applicator is used for the 6 and 18MeV electron beams, and the 10×10/14×14 cm2 applicator is used for the 12MeV electron beams setting a lead plate opening of 10×10 cm^{2} in the 14×14 cm^{2} applicator. The dose studies are classified into three categories: (a) datasets caused by contaminant photons from the treatment head, (b) datasets caused by direct electrons that have not interacted in the electron applicator, and (c) datasets caused by indirect electrons that have interacted in the electron applicator. For each group of the 6, 12, and 18MeV electron beams, the study work is performed using two eMC algorithms by setting a common virtual accelerator; one is that developed by them as the standard eMC, and the other is a commercial treatment planning system (TPS). The present paper refers to these datasets as “the WK eMC dose datasets” or “the WK eMC dose work.”
The eMC treatment is usually a timeconsuming work. Iwasaki and others [3, 4] have reported papers to recalculate the WK eMC dose datasets using an analytical method to shorten the treatment time. However, the reported papers have the following defects:
• Both the depthdose (DD) datasets and the offaxis dose (OAD) datasets were inaccurately calculated in the region near the beam entrance surface by taking unreasonable calculation procedures for calculation of the factor.
• Parallelbeam depthdose datasets (D_{∞}) of infinite field were not precisely estimated at deep depths beyond about the extrapolated range (R_{p}).
• Each set of the OAD profiles was not reconstructed by taking account of the balanced forms at deep depths.
This paper will report how these three problems are overcome for calculating doses more precisely using a mathematical analysis for a simplified homogenous water phantom, preparing future studies how to calculate doses also analytically keeping high accuracy even for heterogeneous phantoms.
MethodsTop
We describe this section based on the electron beam DD and OAD datasets of the WK eMC dose work [2]. As described above, each of the DD and OAD datasets is yielded by setting a common virtual accelerator for both of the standard eMC and the TPS eMC. Here, it should be emphasized that each of the DD and OAD datasets is normalized with a dose of 1.0 Gy per 100 MU at the maximum dose depth (d_{max}) yielded by all particles of contaminant photons, direct electrons, and indirect electrons under the use of an open electron applicator of A_{appl}=20×20 cm^{2}. We use the same dose unit of Gy/100MU for each of the DD and OAD datasets. Here, in this paper, the “no unit” means a constant value when the factor is set exponentially and used for multiplication with the Z_{c}axis in cm.
2.1. DD and OAD datasets
First, we describe how each of the electron beam DD and OAD datasets is expressed using a set of expressions for the direct electron beams and for the directplusindirect electron beams. As listed in Table 1, we use the same KK numbers of KK=1 to 24 for the DD and OAD datasets published in the WK paper of Ref. [2], as originally introduced in the previous papers [3, 4], also indicating the beam energy (E), the electron applicator (A_{appl}), and the standard or TPS eMC. It should be noted that each of the OAD datasets is yielded on two horizontal planes at shallow or deep Zc depth (we specify the two depths as Z_{1} and Z_{2}, respectively), and that, although each DD dataset has no KK number directly, it is used in common with the corresponding two OAD datasets. Namely, the KK numbers also indicate the corresponding diagrams used in the WK paper (Supp. Fig. 3a & 3b, etc.).
Table 1a is constructed for the direct electron beams:
(i) The standard eMC datasets are classified into:
Under the DD dataset in Supp. Fig. 3a, we set KK=1 for Supp. Fig. 3bOAD (Z_{1}=1 cm) and set KK=2 for Supp. Fig. 3cOAD (Z_{2}=5 cm);
Under the DD dataset in Supp. Fig. 5a, we set KK=3 for Supp. Fig. 5bOAD (Z_{1}=2 cm) and set KK=4 for Supp. Fig. 5cOAD (Z_{2}=10 cm);
Under the DD dataset in Supp. Fig. 3d, we set KK=5 for Supp. Fig. 3eOAD (Z_{1}=3 cm) and set KK=6 for Supp. Fig. 3fOAD (Z_{2}=15 cm).
(ii) The TPS eMC datasets are classified into:
Under the DD dataset in Supp. Fig. 3a, we set KK=7 for Supp. Fig. 3bOAD (Z_{1}=1 cm) and set KK=8 for Supp. Fig. 3cOAD (Z_{2}=5 cm);
Under the DD dataset in Supp. Fig. 5a, we set KK=9 for Supp. Fig. 5bOAD (Z_{1}=2 cm) and set KK=10 for Supp. Fig. 5cOAD (Z_{2}=10 cm);
Under the DD dataset in Supp. Fig. 3d, we set KK=11 for Supp. Fig. 3eOAD (Z_{1}=3 cm) and set KK=12 for Supp. Fig. 3fOAD (Z_{2}=15 cm).
Table 1b is constructed for the directplusindirect electron beams:
(i) The standard eMC datasets are classified into:
Under the DD dataset in Supp. Fig. 3a, we set KK=13 for Supp. Fig. 3bOAD (Z_{1}=1 cm) and set KK=14 for Supp. Fig. 3cOAD (Z_{2}=5 cm);
Under the DD dataset in Supp. Fig. 5a, we set KK=15 for Supp. Fig. 5bOAD (Z_{1}=2 cm) and set KK=16 for Supp. Fig. 5cOAD (Z_{2}=10 cm);
Under the DD dataset in Supp. Fig. 3d, we set KK=17 for Supp. Fig. 3eOAD (Z_{1}=3 cm) and set KK=18 for Supp. Fig. 3fOAD (Z_{2}=15 cm).
(ii) The TPS eMC datasets are classified into:
Under the DD dataset in Supp. Fig. 3a, we set KK=19 for Supp. Fig. 3bOAD (Z_{1}=1 cm) and set KK=20 for Supp. Fig. 3cOAD (Z_{2}=5 cm);
Under the DD dataset in Supp. Fig. 5a, we set KK=21 for Supp. Fig. 5bOAD (Z_{1}=2 cm) and set KK=22 for Supp. Fig. 5cOAD (Z_{2}=10 cm);
Under the DD dataset in Supp. Fig. 3d, we set KK=23 for Supp. Fig. 3eOAD (Z_{1}=3 cm) and set KK=24 for Supp. Fig. 3fOAD (Z_{2}=15 cm).
2.2. Effective field side ()
In the previous papers [3, 4], we made mistakes for setting effective fields of at shallow depths of Z_{c} < Z_{1}. Although no large dose errors happen in the DD curves at Z_{c} depths beyond about 1E05 cm for each beam energy of E=6, 12, and 18 MeV, the OAD curve forms gradually narrower widths as the Z_{c} depth decreases from each Z_{1} depth as indicated in Ref. [4]. The present paper proposes another treatment for the field at depths of Z_{c} < Z_{1}. Here, we express the effective field side () for each beam as a function of Z_{c} for a given beam energy (E). We use functional expressions of (Z_{c}) and (Z_{c}, E) case by case.
For the blue lines, as illustrated in Figure 1, we set (0) on the beam entrance surface (Z_{c} = 0), where we simply set the value of (0) as the field side determined using the geometry of the fan beam coming out from the electron applicator by the use of the effective sourcetosurface distance of SSD_{eff} [3]. Then we set (Z_{1}) for the depth of Z_{c} = Z_{1}, and set (Z_{2}) for the depth of Z_{c} = Z_{2}. [we can set (Z_{c}=0) (Z_{c} = Z_{1}) holding (Z_{c} = 0) ]. Table 2 (composed of Tables 2a to 2d) lists datasets for both standard eMC and the TPS eMC. It should be noted that the (0) values are almost constant (10.5 or 10.6 cm) for each irradiation because of a simplified treatment for the determination (it should be noted that the brown line is used by mistake for the evaluation at depths of Z_{c} < Z_{1} in the former papers [3, 4]).
The following expression [3, 4] has been reported for the determination of S_c^eff at depths of Z_{c} ≥ Z_{1} using sets of a_{1}, b_{1}, and c_{1} factors (see Table 1):
For depths of Z_{c} < Z_{1}, this paper proposes the following expression:
where we let (E) be equal to the (0) field side at Zc=0 determined using the abovedescribed geometry of the fan beam coming out from the electron applicator, and then the value of (E) is determined by setting
Table 1 also lists sets of (E) and (E) values for the KK numbers (this paper also uses a simplified expression of (Z_{c}) for (Z_{c}, E) in case of taking a constant beam energy E beforehand). Figure 1 shows how the effective field side () varies with depth (Z_{c}). As described above, Z_{1} and Z_{2} are specified depths used in the OAD datasets of the WK paper (it should be emphasized that we can set (0) (Z_{1}); it seems that the WK paper takes such a special Z_{1} value for each beam energy (E)).
2.3. Parallel beam depthdoses at infinite field
For the dose calculation for a given electron beam, the Gaussian pencil beam model uses a dataset of parallel beam depthdose (D_{∞}) in an infinitely broad field. We have had experiences [3, 4] that the D_{∞} datasets could reasonably be reconstructed for depths less than about the extrapolated range (R_{p}); however, not reasonably be reconstructed for depths beyond about the Rp range. The present paper proposes a reasonable procedure for it as follows:
Figure 2 shows a blue line expressing a raw set of D_{∞} data for a given beam energy E, appearing unreasonable data at depths greater than Z_{c} = Z_{0} (E) taking a dose of a_{0} (E) at depth of Z_{0} (E). Then this paper proposes a different set of D_{∞} data at depths of Z_{c} > Z_{0} (E) using a brown line as
It should be noted that the constant parametervalue of 0.7 is determined by examination of sets of OAD curves produced for all the beam energies of 6, 12, and 18 MeV, and that the value of b_{0} (E) is determined by examination of a set of OAD curves of E under the constant 0.7parameter, especially by taking into account the OAD curve at the depth of Z2. Table 3 lists datasets of a_{0} (E), b_{0} (E), and Z_{0} (E) values (a) for the direct electron beams (KK=112) and (b) for the directplusindirect electron beams (KK=1324).
2.4. Calculation of depthdoses for fan beams
It should be emphasized that the dose calculation for the fan beams is performed using the same procedures as in the former papers of Refs. [3 and 4], only excepting: (i) the mathematical treatment for calculation of the effective field side () for depths of Z_{c} < Z_{1} using Eq. 2 (Figure 1); and (ii) the mathematical treatment for calculation of the parallel beam dose dataset (D_{para}) of infinite field at deep depths using Eq. 4 (Figure 2).
Results and discussionTop
Supplementary Figures (Supp. Fig.) 314 illustrate (a) & (b) DD datasets and (c) & (d) OAD datasets for E=6, 12, and 18 MeV beams, being partly compared with the corresponding dose datasets copied directly from the WK eMC dose work (the standard or TPS eMC). For each DD or OAD dataset, we give a detailed explanation using the factors given in Figure 2 and given in Tables 1 and 2 as follows:
• Supp. Fig. 3 shows the case of the direct electron beams based on the standard eMC for each of the (a)(d) diagrams with respect to KK=1 and 2 (E=6 MeV). It should be noted that the (a) and (b) diagrams express DD datasets, respectively, in a wide Z_{c} region and only in a deep Z_{c} region. In each of the (a) and (b) diagrams, the blue line expresses the doses of calculation, the set of gray marks expresses the doses copied from the WK eMC dose datasets, the yellow line expresses the parallelbeam doses at infinite field, the brawn and blue marks express the doses of infinite field at Z_{c} = Z_{2} and Z_{c} = Z_{0}(E), respectively (Figure 2). On the other hand, in each of the (c) and (d) diagrams, the solid lines express the OAD datasets of calculation, and the two sets of round marks express the OAD datasets copied from the WK eMC dose datasets. It can be seen that the OAD datasets of calculation are reconstructed also by taking account of the balanced forms from shallow to deep depths and also by coinciding well with the WK eMC dose datasets using Eqs. 14. It can be seen that much more accurate DD and OAD curves are yielded even at small Z_{c} regions, when compared with the results in Ref. [4] (this means that the function of Eq. 2 is reasonable).
• Supp. Fig. 4 shows the case of KK=3 and 4 (E=12 MeV) with the direct electron beams based on the standard eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 5 shows the case of KK=5 and 6 (E=18 MeV) with the direct electron beams based on the standard eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 6 shows the case of KK=7 & 8 (E=6 MeV) with the direct electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 7 shows the case of KK=9 & 10 (E=12 MeV) with the direct electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 8 shows the case of KK=11 & 12 (E=18 MeV) with the direct electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 9 shows the case of KK=13 & 14 (E=6 MeV) with the directplusindirect electron beams based on the standard eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 10 shows the case of KK=15 & 16 (E=12 MeV) with the directplusindirect electron beams based on the standard eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 11 shows the case of KK=17 & 18 (E=18 MeV) with the directplusindirect electron beams based on the standard eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 12 shows the case of KK=19 & 20 (E=6 MeV) with the directplusindirect electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 13 shows the case of KK=21 & 22 (E=12 MeV) with the directplusindirect electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
• Supp. Fig. 14 shows the case of KK=23 & 24 (E=18 MeV) with the directplusindirect electron beams based on the TPS eMC. The others are the same as in Supp. Fig. 3.
From the present investigation, the following features have been observed:
Dose differences between the DD curve of finite field and that of infinite field become greater as the increase of the beam energy (E). This means that the range of laterally scattered electrons becomes greater with the increase of the beam energy (E). This phenomenon can be explained by the or function (Refs. 5 and 6).
The revised Gaussianpencilbeammodel uses a mathematical σ_r expression, being reconstructed based on datasets of for E=6,10,14,and 20 MeV as reported by Bruinvis et al. [5].
The three DD points of a_{0} (E) for E=18 MeV in Figs. 8, 11, and 14 are respectively placed at a little wrong places. This fact has been found when writing this paper (we do not dare correct them in this paper).
The weak three points of the former papers of Refs. [3 and 4 regarding the calculated DD and OAD datasets, as noted in the introduction section, have been resolved.
ConclusionTop
To conclude this research report, we would like to conduct research on the development of the Gaussian pencil beam model when the phantom contains nonuniform materials and the beam entrance surface of the phantom is uneven.
Conflicts of interest
This study was carried out in collaboration with Technology of Radiotherapy Corporation, Tokyo, Japan. This sponsor had no control over the interpretation, writing, or publication of this work.
Supplementary dataTop
ReferencesTop
[1]Wieslander E, Knöös T. A virtual linear accelerator for verification of treatment planning systems. Phys Med Biol. 2000; 45:2887–2896.Article Pubmed
[2]Wieslander E, Knöös T. A virtualacceleratorbased verification of a Monte Carlo dose calculation algorithm for electron beam treatment planning in homogeneous phantoms. Phys Med Biol. 2006; 51:1533–1544.Article Pubmed
[3]Iwasaki A, Terashima S, Kimura S, Sutoh K, Kamimura K, et al. A revised Gaussian pencil beam model for calculation of the inwater dose caused by clinical electronbeam irradiation. J Radiol Imaging. 2022; 6:1–9.Article
[4]Iwasaki A, Terashima S, Kimura S, Sutoh K, Kamimura K, et al. Further development of the preceding Gaussianpencilbeammodel used for calculation of the inwater dose caused by clinical electronbeam irradiation. J Radiol Imaging. 2023; 7:1–4.Article
[5]Bruinvis IAD, Amstel AV, Elevelt AJ, Laarse RVD. Calculation of electron beam dose distributions for arbitrarily shaped fields. Phys Med Biol. 1983; 28:667–683.Article Pubmed
[6]Khan FM, Gibbons JP. Khan’s The physics of radiation therapy; 5th edition. Philadelphia, USA; 2014. Available from:.https://solution.lww.com/book/show/446502